Urban Architect Guide Urban Architect Guide
Lesson: The Boundary Blueprint
Subject
Geometry / Pre-Calculus
Learning Objective
Students will construct Voronoi diagrams using geometric principles (perpendicular bisectors) to partition a plane into regions of proximity, applying these techniques to optimize urban facility placement.
Essential Questions
How can we mathematically define "nearest" in a 2D plane?
What geometric properties define the boundary between two points?
How does urban planning use spatial data to ensure equitable access to resources?
Materials
Compass and Straightedge
Blueprint Slides
City Seeders Activity Sheet
Precision Plotting Worksheet
Graph Paper & Colored Pencils
Instructional Sequence
00-10m
The Emergency Room Problem
Display a map of three hospitals in a city. Ask students: "If you are at an accident site, how do you mathematically prove which hospital is the absolute closest?" Discuss the concept of the perpendicular bisector as the equidistant boundary.
10-25m
Direct Instruction: Constructing Boundaries
Use the Blueprint Slides to demonstrate the algorithm:
Identify "Seed Points" (sites).
Connect adjacent sites with dashed segments.
Construct the perpendicular bisector for each segment.
Identify "Voronoi Vertices" where three or more bisectors meet (circumcenters).
25-45m
City Seeders Activity
Hand out the City Seeders Activity . Guide students through constructing a 3-point Voronoi diagram. Monitor for accurate use of the compass and identification of the circumcenter.
45-75m
Precision Plotting & Algebraic Verification
Students work on the Precision Plotting Worksheet . This transitions from manual construction to algebraic calculation (finding equations of lines, midpoints, and intersection points).
75-90m
Metro Masterplan Launch
Introduce the Metro Masterplan Project . This serves as the summative assessment where students design a city layout and justify their placement of fire stations or schools using Voronoi diagrams.
Teacher Tips & Differentiation
Support
Provide pre-calculated midpoints for the worksheet. Use patty paper for folding perpendicular bisectors to visualize the boundaries before drawing.
Challenge
Have students investigate "Weighted Voronoi Diagrams" where some seeds (like larger hospitals) have a larger "pull" or radius of influence.
Blueprint Slides Blueprint Slides
Mapping Space and Proximity with Voronoi Diagrams
URBAN DESIGN
GEOMETRY
The Nearest Neighbor Problem
Imagine you are a city planner placing 3 new hospitals.
"How do we divide the city so every resident goes to the hospital closest to them?"
This isn't just about lines; it's about life-saving efficiency.
City Map: Hospital Sites
What is a Voronoi Diagram?
A Partition of Space
A way of dividing space into regions close to each of a given set of objects (seeds).
The Seeds
The specific points (hospitals, cell towers) that influence the space.
The Cells
The regions where every point is closer to that cell's seed than any other.
The Construction Algorithm
01
Connect
Draw dashed segments between neighboring seeds.
02
Bisect
Construct the perpendicular bisector for each segment. These lines are the boundaries.
03
Intersect
Find the Voronoi Vertices where the bisectors meet. Trim lines to form cells.
Why the Perpendicular Bisector?
Every point on the perpendicular bisector is equidistant from the two seeds.
This line forms the absolute boundary between "closer to A" and "closer to B".
Key Concept
A B
Beyond the Blueprint
Telecom
Optimizing cell tower coverage regions.
Biology
Modeling forest canopy competition.
Marketing
Analyzing trade areas for retail stores.
City Seeders Activity City Seeders Activity
Geometric Construction Lab
Name:
Date:
Challenge Objective
You are tasked with dividing "Grid City" into emergency response zones for three fire stations (the seeds). Every point in a zone must be closer to its station than to any other.
Tools Needed:
Compass
Straightedge
Colored Pencils
Procedure:
Connect seeds with dashed lines.
Construct 3 perpendicular bisectors.
Mark the Voronoi Vertex (the circumcenter).
Station A
Station B
Station C
Blueprint Canvas
Post-Construction Analysis
1. Where the three perpendicular bisectors meet, what is special about that point in relation to the three stations? Use geometric terminology.
2. Pick a random point deep inside "Zone B". If you travel in a straight line toward "Station A", which boundary will you cross first? Explain why.
Precision Plotting Worksheet Precision Plotting Worksheet
Algebraic Foundations of Voronoi Boundaries
Name: ________________________
Perpendicular Bisector Checklist
Find the Midpoint of the segment.
Calculate the Slope of the segment.
Find the Negative Reciprocal slope.
Write the equation in Point-Slope form.
Core Formulas
\(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
\(m_{\perp} = -\frac{1}{m}\)
1 Boundary Between Two Clinics
Clinic A is located at (-4, 2) and Clinic B is at (6, 8). Calculate the equation of the boundary line that separates their regions.
Step 1: Midpoint calculation
Step 2: Calculate Original Slope
Step 3: Perpendicular Slope
Step 4: Final Line Equation
2 The Voronoi Vertex
The boundary line between Clinic B and Clinic C is calculated to be y = -2x + 15. Using your result from Problem 1, find the coordinates of the Voronoi Vertex where these two boundary lines intersect.
Algebraic Work (Show all steps for the system):
Challenge: Proximity Test
If a person is located at (0, 0), which clinic is mathematically closer to them? Justify your answer using the boundary equation from Problem 1.
Metro Masterplan Project Metro Masterplan Project
Summative Performance Task: Urban Spatial Optimization
Project Code
V-092-ALPHA
The Architectural Brief
The City Council of New Geometry has approved 5 major public infrastructure nodes. As the Lead Urban Architect, you must partition the city into service regions. Every citizen must be assigned to the node mathematically closest to them.
Final Deliverables
1
The Technical Blueprint
A full-scale Voronoi diagram on graph paper (min 20x20 units) showing the 5 seeds and their unique service cells.
2
The Boundary Registry
Algebraic equations for at least 3 of your cell boundaries. Show midpoint and slope calculations for each.
3
The Vertex Analysis
Identify one Voronoi vertex (circumcenter) and prove its coordinates by solving the system of its boundary lines.
Design Constraints
Use a straightedge for all lines.
Color-code each Voronoi cell.
Label every vertex and boundary.
Planner's Tip
The boundaries of a Voronoi diagram always consist of line segments, not infinite lines. Trim your perpendicular bisectors once you find where they intersect!
Grading Weight
Accuracy 40%
Algebra 40%
Presentation 20%
Initial Design & Coordinates
Plot your seed coordinates below before starting construction:
Seed 1
( , )
Seed 2
( , )
Seed 3
( , )
Seed 4
( , )
Seed 5
( , )
Rough Sketch / Planning Area:
© Urban Design Board - BB-PROJ-V1.2 Geometry Standards: G-CO.C.10, G-GPE.B.5
